OpenStax-CNX module: m18924
1
Probability Topics: Homework
(modified R. Bloom)
∗
Roberta Bloom
†
Based on Probability Topics: Homework
by
Barbara Illowsky, Ph.D.
Susan Dean
This work is produced by OpenStax-CNX and licensed under the
‡
Creative Commons Attribution License 3.0
Abstract
This module provides a number of homework exercises related to basic concepts and methods in
probability.
This revision of the original module by Dr.
B. Illowsky and S. Dean in the textbook
collection Collaborative Statistics has new problems added at the end of the module.
Exercise 1
(Solution on p. 13.)
Suppose that you have 8 cards. 5 are green and 3 are yellow. The 5 green cards are numbered
1, 2, 3, 4, and 5. The 3 yellow cards are numbered 1, 2, and 3. The cards are well shued. You
randomly draw one card.
• G
• E
= card drawn is green
= card drawn is even-numbered
a.
b. P (G) =
c. P (G|E) =
d. P (G AND E) =
e. P (G OR E) =
f. G E
Exercise 2
with replacement
List the sample space.
Are
and
mutually exclusive? Justify your answer numerically.
Refer to the previous problem. Suppose that this time you randomly draw two cards, one at a
time, and
• G1 =
∗ Version
.
rst card is green
1.3: Nov 15, 2010 3:02 pm -0600
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‡ http://creativecommons.org/licenses/by/3.0/
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OpenStax-CNX module: m18924
• G2 =
2
second card is green
a.
b. P (G
G )=
c. P (
)=
d. P (G | G ) =
e. G G
Exercise 3
without replacement
Draw a tree diagram of the situation.
1 AND
2
at least one green
2
Are
1
2 and
1 independent events? Explain why or why not.
(Solution on p. 13.)
Refer to the previous problems. Suppose that this time you randomly draw two cards, one at a
time, and
.
• G1 =
• G2 =
rst card is green
second card is green
a.
b>. P (G
G ) =
c.
d. P (G |G ) =
e. G G
Exercise 4
Draw a tree diagram of the situation.
1 AND
2
P(at least one green) =
2
Are
1
2 and
1 independent events? Explain why or why not.
Roll two fair dice. Each die has 6 faces.
a.
b.
c.
d.
e.
f.
List the sample space.
A be the event that either a 3 or 4 is rolled rst, followed by an even number. Find P (A).
B be the event that the sum of the two rolls is at most 7. Find P (B).
In words, explain what P (A|B) represents. Find P (A|B).
Are A and B mutually exclusive events? Explain your answer in 1 - 3 complete sentences,
Let
Let
including numerical justication.
Are
A
and
B
independent events? Explain your answer in 1 - 3 complete sentences, including
numerical justication.
Exercise 5
(Solution on p. 13.)
A special deck of cards has 10 cards. Four are green, three are blue, and three are red. When a
card is picked, the color of it is recorded. An experiment consists of rst picking a card and then
tossing a coin.
a.
b.
c.
d.
List the sample space.
Let
A
be the event that a blue card is picked rst, followed by landing a head on the coin toss.
Find P(A).
Let
B
be the event that a red or green is picked, followed by landing a head on the coin toss.
Are the events
A and B
mutually exclusive? Explain your answer in 1 - 3 complete sentences,
including numerical justication.
Let
C
be the event that a red or blue is picked, followed by landing a head on the coin toss.
Are the events
A and C
mutually exclusive? Explain your answer in 1 - 3 complete sentences,
including numerical justication.
Exercise 6
An experiment consists of rst rolling a die and then tossing a coin:
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a.
b.
c.
3
List the sample space.
Let
A
be the event that either a 3 or 4 is rolled rst, followed by landing a head on the coin
toss. Find P(A).
Let
B
be the event that a number less than 2 is rolled, followed by landing a head on the coin
toss.
Are the events
A
and
B
mutually exclusive?
Explain your answer in 1 - 3 complete
sentences, including numerical justication.
Exercise 7
(Solution on p. 13.)
An experiment consists of tossing a nickel, a dime and a quarter. Of interest is the side the coin
lands on.
a.
b.
c.
List the sample space.
Let
Let
A
B
be the event that there are at least two tails. Find P(A).
be the event that the rst and second tosses land on heads.
Are the events
A
and
B
mutually exclusive? Explain your answer in 1 - 3 complete sentences, including justication.
Exercise 8
Consider the following scenario:
•
•
•
= 0.4
= 0.5
P(C|D) = 0.6
Let P(C)
Let P(D)
Let
a.
b. C
c. C
d.
e.
Exercise 9
E
F
Exercise 10
J
K
Exercise 11
Find P(C AND D) .
Are
and
Are
and
D mutually exclusive? Why or why not?
D independent events? Why or why not?
Find P(C OR D) .
Find P(D|C).
U
(Solution on p. 13.)
P (E) = 0.4; P (F ) = 0.5.
and
mutually exclusive events.
and
are independent events. P(J | K)
and
V
Find
P (J)
P (E | F ).
.
(Solution on p. 13.)
are mutually exclusive events.
a.
b.
c.
Exercise 12
Q
R
Exercise 13
= 0.3.
Find
P (U ) = 0.26; P (V ) = 0.37.
Find:
P(U AND V) =
P(U | V)
=
P(U OR V) =
and
Y
and
are independent events.
Z
P (Q) = 0.4 ; P (Q AND R) = 0.1
. Find
P (R).
(Solution on p. 13.)
are independent events.
a.
b.
Exercise 14
Rewrite the basic Addition Rule P(Y OR Z)
= P (Y) + P (Z) − P (Y AND Z)
information that Y and Z are independent events.
Use the rewritten rule to nd
G
and
H
P (Z)
if
are mutually exclusive events.
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P (Y OR Z) = 0.71
and
P (G) = 0.5; P (H) = 0.3
P (Y) = 0.42
.
using the
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a.
b.
c. G
Exercise 15
4
Explain why the following statement MUST be false:
P (H | G) = 0.4
.
Find: P(H OR G).
Are
and
H
independent or dependent events? Explain in a complete sentence.
(Solution on p. 13.)
The following are real data from Santa Clara County, CA. As of March 31, 2000, there was a total
of 3059 documented cases of AIDS in the county. They were grouped into the following categories
(Source: Santa Clara County Public H.D.):
Homosexual/Bisexual IV Drug User* Heterosexual Contact Other Totals
Female
0
70
136
49
____
Male
2146
463
60
135
____
Totals
____
____
____
____
____
Table 1:
* includes homosexual/bisexual IV drug users
Suppose one of the persons with AIDS in Santa Clara County is randomly selected. Compute
the following:
a.
b.
c.
d.
e.
f.
g.
Exercise 16
P(person is female) =
P(person has a risk factor Heterosexual Contact) =
P(person is female OR has a risk factor of IV Drug User) =
P(person is female AND has a risk factor of Homosexual/Bisexual) =
P(person is male AND has a risk factor of IV Drug User) =
P(female GIVEN person got the disease from heterosexual contact) =
Construct a Venn Diagram. Make one group females and the other group heterosexual contact.
Solve these questions using probability rules.
Do NOT use the contingency table above.
3059
cases of AIDS had been reported in Santa Clara County, CA, through March 31, 2000. Those cases
will be our population.
Of those cases, 6.4% obtained the disease through heterosexual contact
and 7.4% are female. Out of the females with the disease, 53.3% got the disease from heterosexual
contact.
a.
b.
c.
d.
P(person is female) =
P(person obtained the disease through heterosexual contact) =
P(female GIVEN person got the disease from heterosexual contact) =
Construct a Venn Diagram. Make one group females and the other group heterosexual contact.
Fill in all values as probabilities.
Exercise 17
(Solution on p. 14.)
The following table identies a group of children by one of four hair colors, and by type of hair.
Hair Type Brown Blond Black Red Totals
Wavy
20
Straight
80
Totals
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15
15
20
3
43
12
215
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5
Table 2
a.
b.
c.
d.
e.
f.
g.
Exercise 18
boys
Complete the table above.
What is the probability that a randomly selected child will have wavy hair?
What is the probability that a randomly selected child will have either brown or blond hair?
What is the probability that a randomly selected child will have wavy brown hair?
What is the probability that a randomly selected child will have red hair, given that he has
straight hair?
If B is the event of a child having brown hair, nd the probability of the complement of B.
In words, what does the complement of B represent?
A previous year, the weights of the members of the
San Francisco 49ers
were published in the San Jose Mercury News.
and the
Dallas Cow-
The factual data are compiled into the
following table.
Shirt#
≤
210 211-250 251-290 290≤
1-33
21
5
0
0
34-66
6
18
7
4
66-99
6
12
22
5
Table 3
For the following, suppose that you randomly select one player from the 49ers or Cowboys.
a.
b.
c.
d.
e.
f.
Find the probability that his shirt number is from 1 to 33.
Find the probability that he weighs at most 210 pounds.
Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds.
Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210 pounds.
Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at most 210
pounds.
If having a shirt number from 1 to 33 and weighing at most 210 pounds were independent events,
then what should be true about P(Shirt# 1-33 |
≤
210 pounds)?
Exercise 19
(Solution on p. 14.)
Approximately 249,000,000 people live in the United States.
Of these people, 31,800,000 speak
a language other than English at home. Of those who speak another language at home, over 50
percent speak Spanish. (Source: U.S. Bureau of the Census, 1990 Census )
Let:
E
= speak English at home;
E'
= speak another language at home;
S
= speak Spanish at
home
Finish each probability statement by matching the correct answer.
Probability Statements
Answers
continued on next page
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a. P(E') =
i. 0.8723
b. P(E) =
ii.
c. P(S) =
iii. 0.1277
d. P(S|E') =
iv.
>
0.50
>
0.0639
Table 4
Exercise 20
The probability that a male develops some form of cancer in his lifetime is 0.4567 (Source:
American Cancer Society). The probability that a male has at least one false positive test result
(meaning the test comes back for cancer when the man does not have it) is 0.51 (Source: USA
Today).
Some of the questions below do not have enough information for you to answer them.
Write not enough information for those answers.
Let:
C
= a man develops cancer in his lifetime;
a.
b. P (C)
c. P (P|C)
d. P (P|C )
e.
P
= man has at least one false positive
Construct a tree diagram of the situation.
=
=
'
=
If a test comes up positive, based upon numerical values, can you assume that man has cancer?
Justify numerically and explain why or why not.
Exercise 21
(Solution on p. 14.)
In 1994, the U.S. government held a lottery to issue 55,000 Green Cards (permits for non-citizens
to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately 6.5 million
people who entered this lottery. Let G = won Green Card.
a.
b.
What was Renate's chance of winning a Green Card?
Write your answer as a probability
statement.
In the summer of 1994, Renate received a letter stating she was one of 110,000 nalists chosen.
Once the nalists were chosen, assuming that each nalist had an equal chance to win, what
was Renate's chance of winning a Green Card? Let F = was a nalist. Write your answer as
c.
d.
a conditional probability statement.
Are
G and F
independent or dependent events? Justify your answer numerically and also explain
why.
Are
G
and
F
mutually exclusive events? Justify your answer numerically and also explain why.
note: P.S. Amazingly, on 2/1/95, Renate learned that she would receive her Green Card true
story!
Exercise 22
Three professors at George Washington University did an experiment to determine if economists
are more selsh than other people. They dropped 64 stamped, addressed envelopes with $10 cash
in dierent classrooms on the George Washington campus. 44% were returned overall. From the
economics classes 56% of the envelopes were returned. From the business, psychology, and history
classes 31% were returned. (Source: Wall Street Journal )
Let:
a.
b.
R
= money returned;
E
= economics classes;
O
= other classes
Write a probability statement for the overall percent of money returned.
Write a probability statement for the percent of money returned out of the economics classes.
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c.
d.
e.
7
Write a probability statement for the percent of money returned out of the other classes.
Is money being returned independent of the class? Justify your answer numerically and explain
it.
Based upon this study, do you think that economists are more selsh than other people? Explain
why or why not. Include numbers to justify your answer.
Exercise 23
(Solution on p. 14.)
The chart below gives the number of suicides estimated in the U.S. for a recent year by age, race
(black and white), and sex. We are interested in possible relationships between age, race, and sex.
We will let suicide victims be our population. (Source: The National Center for Health Statistics,
U.S. Dept. of Health and Human Services )
Race and Sex 1 - 14 15 - 24 25 - 64 over 64 TOTALS
white, male
210
3360
13,610
22,050
white, female
80
580
3380
4930
black, male
10
460
1060
1670
black, female
0
40
270
330
310
4650
18,780
29,760
all others
TOTALS
Table 5
note: Do not include "all others" for parts (f ), (g), and (i).
a.
b.
c.
d.
e.
f.
g.
h.
i.
The next two questions refer to the following:
Fill in the column for the suicides for individuals over age 64.
Fill in the row for all other races.
Find the probability that a randomly selected individual was a white male.
Find the probability that a randomly selected individual was a black female.
Find the probability that a randomly selected individual was black
Comparing Race and Sex to Age, which two groups are mutually exclusive?
How do you
know?
Find the probability that a randomly selected individual was male.
Out of the individuals over age 64, nd the probability that a randomly selected individual was
a black or white male.
Are being male and committing suicide over age 64 independent events? How do you know?
The percent of licensed U.S. drivers (from a recent
year) that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are age 20 - 64; 13.61%
are age 65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and under; 81.43% are age 20 - 64;
13.53% are age 65 or over. (Source: Federal Highway Administration, U.S. Dept. of Transportation)
Exercise 24
Complete the following:
a.
b.
c.
d.
Construct a table or a tree diagram of the situation.
P(driver is female) =
P(driver is age 65 or over | driver is female) =
P(driver is age 65 or over AND female) =
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8
e.
f.
g.
h.
Exercise 25
In words, explain the dierence between the probabilities in part (c) and part (d).
P(driver is age 65 or over) =
Are being age 65 or over and being female mutually exclusive events? How do you know
P(driver is "male" OR "age 19 or under") =
(Solution on p. 14.)
Suppose that 10,000 U.S. licensed drivers are randomly selected.
a.
b.
c.
How many would you expect to be male?
Using the table or tree diagram from the previous exercise, construct a contingency table of
gender versus age group.
Using the contingency table, nd the probability that out of the age 20 - 64 group, a randomly
selected driver is female.
Exercise 26
Approximately 86.5% of Americans commute to work by car, truck or van. Out of that group, 84.6%
drive alone and 15.4% drive in a carpool. Approximately 3.9% walk to work and approximately 5.3%
take public transportation. (Source: Bureau of the Census, U.S. Dept. of Commerce. Disregard
rounding approximations.)
a.
b.
c.
d.
e.
Exercise 27
Construct a table or a tree diagram of the situation. Include a branch for all other modes of
transportation to work.
Assuming that the walkers walk alone, what percent of all commuters travel alone to work?
Suppose that 1000 workers are randomly selected. How many would you expect to travel alone
to work?
Suppose that 1000 workers are randomly selected. How many would you expect to drive in a
carpool?
What percent of workers do NOT "drive alone"?
Explain what is wrong with the following statements. Use complete sentences.
a.
b.
If there's a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then there's a
130% chance of rain over the weekend.
The probability that a baseball player hits a home run is greater than the probability that he
gets a successful hit.
1 Questions 28 through 32 are multiple choice
Questions 28 and 29 refer to the following probability tree diagram
FOLLOWED BY
R
coin
beads. For the coin,
drawing one bead from a cup containing 3 red (
P (H) =
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2
3 and
P (T ) =
which shows tossing an unfair
), 4 yellow (Y ) and 5 blue (B )
1
3 where H = "heads" and T = "tails .
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9
Figure 1
Exercise 28
(Solution on p. 14.)
A.
B.
C.
D.
Exercise 29
(Solution on p. 14.)
Find P(tossing a Head on the coin AND a Red bead)
2
3
5
15
6
36
5
36
Find P(Blue bead).
A.
B.
C.
D.
Questions 30 through 32 refer to the following table
15
36
10
36
10
12
6
36
of data obtained from www.baseball-almanac.com
showing hit information for 4 well known baseball players.
NAME
Single Double Triple Home Run TOTAL HITS
Babe Ruth
1517
506
136
714
2873
Jackie Robinson
1054
273
54
137
1518
Ty Cobb
3603
174
295
114
4189
Hank Aaron
2294
624
98
755
3771
TOTAL
8471
1577
583
1720
12351
Table 6
1 http://cnx.org/content/m18924/latest/www.baseball-almanac.com
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1
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10
Exercise 30
(Solution on p. 14.)
A.
B.
C.
D.
Exercise 31
(Solution on p. 14.)
A.
B.
C.
D.
Exercise 32
(Solution on p. 14.)
Find P(hit was made by Babe Ruth).
1518
2873
2873
12351
583
12351
4189
12351
Find P(hit was made by Ty Cobb | The hit was a Home Run)
4189
12351
1141
1720
1720
4189
114
12351
Are the hit being made by Hank Aaron and the hit being a double independent events?
A.
B.
C.
D.
Exercise 33
Yes, because P(hit by Hank Aaron | hit is a double) = P(hit by Hank Aaron)
No, because P(hit by Hank Aaron | hit is a double)
6=
No, because P(hit is by Hank Aaron | hit is a double)
P(hit is a double)
6=
P(hit by Hank Aaron)
Yes, because P(hit is by Hank Aaron | hit is a double) = P(hit is a double)
(Solution on p. 14.)
Given events G and H: P(G) = 0.43 ; P(H) = 0.26 ; P(H and G) = 0.14
a.
b.
c.
Exercise 34
(Solution on p. 15.)
a.
b.
c.
Exercise 35
(Solution on p. 15.)
Find P(H or G)
Find the probability of the complement of event (H and G)
Find the probability of the complement of event (H or G)
Given events J and K: P(J) = 0.18 ; P(K) = 0.37 ; P(J or K) = 0.45
Find P(J and K)
Find the probability of the complement of event (J and K)
Find the probability of the complement of event (J or K)
United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. According
to their website, http://www.unitedbloodservices.org/humanbloodtypes.html, a person with type
O blood and a negative Rh factor (Rh−) can donate blood to any person with any bloodtype.
Their data show that 43% of people have type O blood and 15% of people have Rh− factor; 52%
of people have type O or Rh− factor.
a.
b.
Exercise 36
Find the probability that a person has both type O blood and the Rh− factor
Find the probability that a person does NOT have both type O blood and the Rh− factor.
(Solution on p. 15.)
At a college, 72% of courses have nal exams and 46% of courses require research papers. Suppose
that 32% of courses have a research paper and a nal exam. Let F be the event that a course has
a nal exam. Let R be the event that a course requires a research paper.
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a.
b.
Exercise 37
11
Find the probability that a course has a nal exam or a research project.
Find the probability that a course has NEITHER of these two requirements.
(Solution on p. 15.)
In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. Of those, 8% contain
both chocolate and nuts. Sean is allergic to both chocolate and nuts.
a.
b.
Exercise 38
Find the probability that a cookie contains chocolate or nuts (he can't eat it).
Find the probability that a cookie does not contain chocolate or nuts (he can eat it).
(Solution on p. 15.)
A college nds that 10% of students have taken a distance learning class and that 40% of students
are part time students. Of the part time students, 20% have taken a distance learning class. Let D
= event that a student takes a distance learning class and E = event that a student is a part time
student
a.
b.
c.
d.
e.
Exercise 39
Find P(D and E)
Find P(E | D)
Find P(D or E)
Using an appropriate test, show whether D and E are independent.
Using an appropriate test, show whether D and E are mutually exclusive.
(Solution on p. 15.)
At a certain store the manager has determined that 30% of customers pay cash and 70% of
customers pay by debit card. (No other method of payment is accepted.) Let M = event that a
customer pays cash and D= event that a customer pays by debit card.
a.
b.
c.
d.
e.
Suppose two customers (Al and Betty) come to the store. Explain why it would be reasonable
to assume that their choices of payment methods are independent of each other.
Draw the tree that represents the all possibilities for the 2 customers and their methods of
payment. Write the probabilities along each branch of the tree.
For each complete path through the tree, write the event it represents and nd the probability.
Let S be the event that both customers use the same method of payment. Find P(S)
Let T be the event that both customers use dierent methods of payment. Find P(T) by two
dierent methods: by using the complement rule and by using the branches of the tree. Your
answers should be the same with both methods.
f.
Exercise 40
Let U be the event that the second customer uses a debit card. Find P(U)
(Solution on p. 15.)
A box of cookies contains 3 chocolate and 7 butter cookies. Miguel randomly selects a cookie and
eats it. Then he randomly selects another cookie and eats it also. (How many cookies did he take?)
a.
b.
c.
d.
e.
Are the probabilities for the avor of the SECOND cookie that Miguel selects independent of
his rst selection, or do the probabilities depend on the type of cookie that Miguel selected
rst? Explain.
Draw the tree that represents the possibilities for the cookie selections. Write the probabilities
along each branch of the tree.
For each complete path through the tree, write the event it represents and nd the probabilities.
Let S be the event that both cookies selected were the same avor. Find P(S).
Let T be the event that both cookies selected were dierent avors. Find P(T) by two dierent
methods: by using the complement rule and by using the branches of the tree. Your answers
should be the same with both methods.
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f.
Exercise 41
12
Let U be the event that the second cookie selected is a butter cookie. Find P(U).
(Solution on p. 15.)
When the Euro coin was introduced in 2002, two math professors had their statistics students test
whether the Belgian 1 Euro coin was a fair coin. They spun the coin rather than tossing it, and it
was found that out of 250 spins, 140 showed a head (event H) while 110 showed a tail (event T).
Therefore, they claim that this is not a fair coin.
a.
b.
c.
d.
Based on the data above, nd P(H) and P(T).
Use a tree to nd the probabilities of each possible outcome for the experiment of tossing the
coin twice.
Use the tree to nd the probability of obtaining exactly one head in two tosses of the coin.
Use the tree to nd the probability of obtaining at least one head.
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13
Solutions to Exercises in this Module
Solution to Exercise (p. 1)
a. {G1, G2, G3, G4, G5, Y1, Y2, Y3}
b.
c.
d.
e.
f.
Solution to Exercise (p. 2)
b. c.
+
+
d.
e.
Solution to Exercise (p. 2)
a. { , , , , , }
b.
c.
d.
Solution to Exercise (p. 3)
a. {( ) , ( ) , ( ) , ( ) , (
b.
c.
Solution to Exercise (p. 3)
Solution to Exercise (p. 3)
a.
b.
c.
Solution to Exercise (p. 3)
b.
Solution to Exercise (p. 4)
5
8
2
3
2
8
6
8
No
5
8
5
8
4
7
No
4
7
3
7
3
8
5
7
5
8
4
7
GH GT BH BT RH RT
3
20
Yes
No
HHH
4
8
Yes
HHT
HTH
HTT
THH) , (THT) , (TTH) , (TTT)}
0
0
0
0.63
0.5
The completed contingency table is as follows:
Female
Male
Totals
Homosexual/Bisexual IV Drug User* Heterosexual Contact Other Totals
255
2804
2146
533
196
184 3059
0
70
136
49
2146
463
60
135
Table 7:
http://cnx.org/content/m18924/1.3/
* includes homosexual/bisexual IV drug users
OpenStax-CNX module: m18924
a.
b.
c.
d.
e.
f.
Solution to Exercise (p. 4)
b.
c.
d.
e.
f.
Solution to Exercise (p. 5)
a.
b.
c.
d.
Solution to Exercise (p. 6)
a. P (G) = 0
b.
c.
d.
Solution to Exercise (p. 7)
c.
d.
e.
f.
g.
h.
i.
Solution to Exercise (p. 8)
a.
c.
Solution to Exercise (p. 9)
Solution to Exercise (p. 9)
Solution to Exercise (p. 10)
Solution to Exercise (p. 10)
Solution to Exercise (p. 10)
Solution to Exercise (p. 10)
255
3059
196
3059
718
3059
0
463
3059
136
196
43
215
120
215
20
215
12
172
115
215
iii
i
iv
ii
.008
0.5
dependent
No
22050
29760
330
29760
2000
29760
23720
29760
5010
6020
Black females and ages 1-14
No
5140
0.49
C
A
B
B
C
http://cnx.org/content/m18924/1.3/
14
OpenStax-CNX module: m18924
15
a.
−
b.
−
c.
−
Solution to Exercise (p. 10)
a.
−
b.
−
c.
−
Solution to Exercise (p. 10)
a.
−
a.
−
a.
−
b.
−
b.
Solution to Exercise (p. 10)
a.
−
b.
−
Solution to Exercise (p. 11)
P(H or G) = P(H) + P(G)
P(H and G) = 0.26 + 0.43
P( NOT (H and G) ) = 1
−
P(H and G) = 1
P( NOT (H or G) ) = 1
P(H or G) = 1
P(J or K) = P(J) + P(K)
−
−
0.14 = 0.55
0.14 = 0.86
0.55 = 0.45
P(J and K); 0.45 = 0.18 + 0.37
−
P(J and K) ; solve to nd P(J and K) =
0.10
P( NOT (J and K) ) = 1
P( NOT (J or K) ) = 1
P(Type O or Rh
P(J and K) = 1
P(J or K) = 1
−
) = P(Type O) + P(Rh−)
0.52 = 0.43 + 0.15
−
0.10 = 0.90
0.45 = 0.55
−
P(Type O and Rh−)
P(Type O and Rh−); solve to nd P(Type O and Rh−) = 0.06
6% of people have type O Rh
P( NOT (Type O and Rh
blood
) ) = 1
−
P(Type O and Rh−) = 1
−
0.06 = 0.94
94% of people do not have type O Rh− blood
P(R or F) = P(R) + P(F)
P( Neither R nor F ) = 1
P(R and F) = 0.72 + 0.46
P(R or F) = 1
−
−
0.32 = 0.86
0.86 = 0.14
Let C be the event that the cookie contains chocolate. Let N be the event that the cookie contains nuts.
a.
−
b.
Solution to Exercise (p. 11)
a.
b.
c.
−
d.
e.
P(C or N) = P(C) + P(N)
P(C and N) = 0.36 + 0.12
P( neither chocolate nor nuts) = 1
−
P(C or N) = 1
−
−
0.08 = 0.40
0.40 = 0.60
P(D and E) = P(D|E)P(E) = (0.20)(0.40) = 0.08
P(E|D) = P(D and E) / P(D) = 0.08/0.10 = 0.80
P(D or E) = P(D) + P(E)
P(D and E) = 0.10 + 0.40
−
0.08 = 0.42
Not Independent: P(D|E) = 0.20 which does not equal P(D) = .10
Not Mutually Exclusive: P(D and E) = 0.08 ; if they were mutually exclusive then we would need to
have P(D and E) = 0, which is not true here.
Solution to Exercise (p. 11)
Solution to Exercise (p. 11)
Solution to Exercise (p. 12)
Solution is posted on instructor's website for this class.
Solution is posted on instructor's website for this class.
Solution is posted on instructor's website for this class.
http://cnx.org/content/m18924/1.3/